A Convergent Adaptive Finite Element Method for Electrical Impedance Tomography

TL;DR

This paper introduces an adaptive finite element method for electrical impedance tomography, improving solution accuracy for this ill-posed inverse problem through a novel error estimator and convergence analysis.

Contribution

It develops a new adaptive algorithm with an a posteriori error estimator for electrical impedance tomography, and proves its convergence.

Findings

Algorithm converges to a solution of the continuous problem.

Numerical results verify convergence and efficiency.

Adaptive method improves reconstruction accuracy.

Abstract

In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage measurements. The reconstruction technique is based on Tikhonov regularization with a Sobolev smoothness penalty and discretizing the forward model using continuous piecewise linear finite elements. We derive an adaptive finite element algorithm with an a posteriori error estimator involving the concerned state and adjoint variables and the recovered conductivity. The convergence of the algorithm is established, in the sense that the sequence of discrete solutions contains a convergent subsequence to a solution of the optimality system for the continuous formulation. Numerical results are presented to verify the convergence and efficiency of the algorithm.